The studied region is defined by 8 × 8 TRMM cells from the southern Peruvian Andes (Fig. ), with a spatial resolution of approximately 28 km. The majority of these TRMM cells are placed in the Peruvian Andean high plateau or Altiplano and a few cells over the east side of the Andes. Geographical coordinates of the study area are between latitudes 14  and 16∘ S and longitudes 69  and 71∘ W constituting an area of approximately 225 × 225 km2. The altitudes range between 800 and 6500 m a.s.l., approximately. The annual rainfall varies, on average, from ∼ 250 mm in the arid southwest to ∼ 5000 mm in the Amazon basin at the northeast corner of the study site . The year to year variability with respect to extreme phenomena, such as El Niño, is mitigated by the Andes and therefore the region's climate can be considered as yearly invariant . The rainfall over the Altiplano is largely concentrated in the austral summer months, when more than 70 % of the precipitation occurs from December to February. On all timescales, the climatic conditions on the Altiplano are closely related to the upper-air circulation, with an easterly zonal flow aloft favoring wet conditions and westerly flow causing dry conditions . At a diurnal timescale, convective clouds are particularly common during the afternoon and evening , and are produced by the insolation-driven surface heating and the consequent destabilization of the local lower troposphere . At intraseasonal timescales, within the rainy season, the Altiplano experiences rainy and dry episodes lasting between 5 and 10 days , as a reflection of the position and intensity of the Bolivian High , which is modulated by Rossby waves emanating from the midlatitude South Pacific. Interannual variability is primarily related to changes in the mean zonal flow over the Altiplano, reflecting changes in meridional baroclinicity between tropical and subtropical latitudes, which in turn is a response to sea-surface temperature changes in the tropical Pacific .

In this area of study, there are 19 stations (black dots in Fig. ) from which observed rainfall measurements are obtained during a period of 8 years (from 1999 to 2006). A tipping bucket rain gauge was used to obtain these on-site rainfall measurements . The role of the data from the stations in this paper is twofold. The first one is to validate the generation of downscaled rainfall data, and the second is to aid in the generation of the deterministic local heterogeneity; see Sect. .

The Tropical Rainfall Measuring Mission is a joint space mission between the National Aeronautics and Space Administration (NASA) and Japan's National Space Development Agency (NASDA) designed to monitor and study tropical and subtropical precipitation and the associated release of energy. The primary rainfall instruments on TRMM are the TRMM Microwave Imager (TMI), the PR (polarization radiometers) and the Visible and Infrared Radiometer System (VIRS). In addition, TRMM satellites carry two related Earth Observing System (EOS) instruments: the Clouds and the Earth's Radiant Energy System (CERES) and the Lightning Imaging Sensor (LIS) . The TMI is the main instrument used for precipitation; see http://climatedataguide.ucar.edu/guidance/trmm-tropical-rainfall-measuring-mission for additional information about the TRMM mission. Our interest is in daily TRMM 3B42 v7 grid data. The information was downloaded directly from http://iridl.ldeo.columbia.edu/SOURCES/.NASA/.GES-DAAC/.TRMM_L3/.TRMM_3B42/.v7/ .daily/ for the period of 1 January 1999–31 December 2006. TRMM 3B42 v7 is a 3 h average, extrapolated to a daily temporal scale, 0.25∘ spatial resolution (∼ 27.8 km) TRMM product derived from calibrated geosynchronous IR imagery merged with TRMM and other satellite data. Hereafter, TRMM 3B42 v7 will be called simply TRMM. TRMM rainfall (mm) is distributed uniformly in each grid cell. A total of 8 × 8 TRMM cells cover the study area with rainfall between 450 and 2000 mm yr-1. Although TRMM tends to underestimate rainfall in terrain with complex topography, we used the correction protocol described in . To correct TRMM daily rainfall over the Andean plateau, the procedure in incorporates the high-frequency component (detail or noise) from rain gauge signals into the low-frequency component (tendency or base) derived from TRMM using wavelet multiresolution analysis (MRA). For each TRMM cell, the high-frequency component of the closest meteorological station was added to the low-frequency component of TRMM. The MRA reconstruction was started at the third level of the Haar wavelet decomposition of TRMM and rain gauge signals. It is important to emphasize that the main reason for selecting the study site was the rainfall heterogeneity found in that small area including arid, semi-arid, and humid tropics and the availability of a rain gauges (Table )

The Tambopata station was not used to correct TRMM rainfall information since there were not enough neighboring stations to accurately represent the details in the region surrounding the station .

Another fact about the corrected TRMM information is that its spatial distribution fits approximately a lognormal distribution, which can be confirmed by a simple lognormality test. Such test is applied to the corrected TRMM by eliminating the zero intensity rainfall and computing its logarithm. As a result, the logarithm of the information mostly agrees with a normal distribution and thus any test for normality should give the desired answer. This procedure must be applied on a daily basis since the parameters of the distribution vary with time. Figure  exemplifies the method for 15 April 2003 and provides evidence of lognormality. The lognormality assumption is also supported, for example, by and . The interested reader can find another perspective on the nature of satellite information and its relationship to the cascade structure of precipitation in .

As the reader can observe in Fig. , there are some discrepancies with respect to the high values of rainfall intensity. There are several factors that may contribute to such discrepancies. First, the northeast corner of the studied region lacks enough meteorological stations to correctly perform the MRA correction of TRMM. That region is in the rain forest and its rainfall variability differs greatly with respect to the rest of the studied area (there is an abrupt change of topography, also the Andes is a high elevation area whereas the rain forest is at a low elevation). Also, each snapshot is comprised of 64 pixels from which the zero values must be removed to perform the test due to the fact that zero is not included in the domain of lognormal distributions. Even though this is a small sample of points for applying an statistical test, this very simple test shows a good relationship between the rainfall field and a lognormal distribution, but we argue that there are not enough extreme values (corresponding to the tail of the distribution), which contributes to the discrepancies shown in Fig. .

To describe a general disaggregation model, it is first assumed that the initial information, at level 0, is comprised by only one value of TRMM rainfall rate (Fig. ). That is, the rainfall total rate ρ0 is uniformly spread over the initial area l02. At level 1, the initial area is partitioned into four identical boxes of length l0/21. For subdivision i at level n, a part of the initial rainfall rate ρ0, denoted as ρni, is uniformly distributed over an area of l02/4n, and is expressed in terms of ρ0 as ρni=ρ0∏j=1nwi,j, where wij are the weights at which the rainfall rate is disaggregated throughout the cascade process. In particular, these weights are outcomes of a random variable W that follows some distribution law and satisfies E[W] = 1, where E[⋅] denotes the expected value of a random variable. This condition also means that if n tends to infinity then ρ∞ will exist and not be degenerated; i.e., ρ∞ ≠ 0. The procedure is repeated until the nth level is reached. After n cascade steps, the original information is disaggregated into 4n pieces.

A good disaggregation model is necessary for a realistic and accurate distribution of the information. This is done by describing the weights w in a probabilistic manner. The class of log-stable random variables constitutes a broad class of generators used for this purpose and therefore characterizes the class of multifractal fields that can be generated; see for an accessible introduction to general multifractal analysis. In particular, such a log-stable generator has the form Y=bγ+σX, where X is a stable (Lévy) random variable . The goal of this paper is to utilize a particular type of such generators. That is, those whose generator is lognormal. In the language of log-stable generators, this corresponds to those characterized by the stability index α = 2 . In order to preserve E[Y] = 1, it follows that E[Y]=Ebγ+σX=bγEbσX=bγEeσXln⁡(b), where the expectation in the right hand side is simply the moment generating function of an unitary normal distribution. Thus, eγln⁡(b)+12σ2ln⁡(b)2 = 1, which implies γ = -σ2ln⁡(b)2 and Y=b-σ2ln⁡(b)2+σX. It is obvious that the generator Y is not able to generate zeros as expected from a precipitation simulation. In this regard, a more realistic generator, W, is given by including an atom at zero in a composition manner (representing the presence and absence of data). That is, W=BY, where P[B=0]=1-b-βandPB=bβ=b-β and is known as the β model. This composition results in the β-lognormal model specified by P[W=0]=1-b-βandPW=bβ-σ2log⁡(b)2+σX=b-β, where X is a standard normal variable, and β and σ fully describe the generator W probability distribution . Other models, in one dimension, that introduce a measure of zero rainfall can be found in and . Since β and σ completely characterize the downscaling process, the key for the generation of a downscaling rainfall model is to estimate these parameters via a scale invariant method. In , and , it was shown that the parameters β and σ can be easily computed using multifractal sample moments obtained directly from rainfall information. Specifically, the multifractal sample moments are defined as Mn(q)=∑i=1bnμniq, where μni = b-nρni is the value of the field at the ith box at the resolution scale λn = ln/l0 with ln being the scale of interest, l0 is the current scale, n ≫ 1 and Mn(q) > 0. To the multifractal moments in Eq. (), one can associate λn to a parameter τ as follows: Mn(q)∝λn-τ(q), which provides τ(q)=lim⁡n→∞log⁡Mn(q)-log⁡λn. An underlying assumption is that the measures of rainfall are independent and identically distributed (iid), and that there exists ergodicity in the rainfall process (at least approximately), which allows us to approximate E[Mn(q)] by Mn(q) .

The multifractal parameter τ characterizes the downscaling process and has the property of being scale invariant . The disaggregation process is then characterized by computing τ for all desired levels for each multifractal moment q. This characterization is analogous to the one provided by the parameters β and σ in Eq. (). Therefore, there exists a relationship between τ and the parameters β and σ. To see such relationship, one has to rely on the MKP function . The MKP function is literally the slope of the linear regression of the log–log plot of Mn(q) versus λn for different values of q, and it is specifically given by χb(q)=log⁡bEWq-(q-1). Theorem 3.2 in adapted a theorem of to account for the presence of zero rainfall giving the relationship τ(q)=dχb(q), where again d constitutes the domain of the cascade (here d = 2). Thus, one can explicitly compute the MKP function for the generator W and equate it with the multifractal sample q moments. That is, χb(q)=log⁡bEWq-(q-1)=1-q+log⁡bE[W=0]+EWq=BqYq=1-q+log⁡bEBq+log⁡bEYq, where EBq=bqβPB=bβ=bβ(q-1) and EYq=b-qσ2log⁡b2EbqσX=b-qσ2log⁡b2bq2σ2log⁡b2=bσ2log⁡b2q2-q. Thus, χb(q)=(β-1)(q-1)+σ2log⁡b2q2-q. By considering the first and second derivatives of χb(q) in Eq. () with respect to the moment q, the parameters β and σ can be obtained as follows: β=1+τ′(q)d-σ2log⁡(b)2(2q-1)andσ2=τ′′(q)dlog⁡(b). The validity of Eq. () is given by the conditions in Theorem 3.2 in ; however, one can empirically test the linearity of log⁡Mn(q) vs. log⁡λn directly from measurements coming from a multifractal field. Thus, the formalism above is valid for the q moments in which linearity is observed. The first and second derivatives of τ with respect to q can be computed, for example, by using finite differences. It is normal to compute the parameters of the β-lognormal model using q = 1. Note that τ(1) is always zero, and this obviously holds for all multifractal fields analyzed in this paper's application. This provides a common point of reference when computing the parameters β and σ. One way to argue the use of q = 1 is that it is related to the mean of the generator, and this is what is preserved at every step of the disaggregation process.

The random generator model is only good for an isotropic distribution of the information at every step of the disaggregation process. From the spatial point of view, the presence of highly variable mountainous terrain in the Andean high plateau adds spatial heterogeneity to any rainfall measurement. From the temporal point of view, averaging rainfall over a given timescale, say hourly or daily, decreases randomness in the sense that averaging acts as a smoothing filter for the rainfall time series. The rainfall field after averaging shows the tendency of the field (usually dependent on the topography of the area), which reveals the heterogeneity in the data due to rainfall intensity. This was originally considered in . In meteorological wording, convective activity from cumulus clouds is much more heterogeneous that the total amount of precipitation observed over a month or larger timescale. Even temporal aggregation on a minute scale to obtain an hourly scale can cause spatial heterogeneity, however, this is usually masked by the information variability. For instance, the accumulated rainfall over all seasons in the Andean high plateau may cause the winter and summer to compensate each other to a large degree and produce a more uniform field than the winter or summer rainfall being measured separately. Since direct downscaling produces homogeneous multifractal fields (each TRMM cell is replaced by a set of random numbers distributed according to the β-lognormal model), the process of the multifractal downscaling model (as any other type of statistical downscaling model) is not enough to keep the heterogeneity information through the downscaling process. , and have applied suitable (pointwise) filters that highlight the effect of spatial heterogeneity. In this respect, rainfall is considered a combined effect of two processes: (1) a multifractal process which is highly variable in space, at least at regional and smaller scales, but statistically uniform; and (2) a deterministic process that represents the heterogeneity of rainfall in space and modifies the multifractal process. This is why prior to applying the β-lognormal model to the TRMM data described in Sect. , it is first necessary to remove the heterogeneity from the TRMM rainfall field. The rainfall field (spatial) under study is denoted as R. R is assumed to have the following decomposition: Ri,j = Mi,jGi,j, with M being a homogeneous multifractal field and G the deterministic component of weights corresponding to long-term monthly averages. For the application of multifractal downscaling, one requires the multifractal field M, which is obtained simply as Mi,j=Ri,jGi,j, where Mi,j = 0 if Gi,j = 0. Specifically, G at position (i, j) is computed as Gi,j=NR‾i,j∑i,jR‾i,j, where R‾i,j is the monthly mean for the pixel location (i, j), and N is the total number of pixels in a daily snapshot. In Sect. , the effect of extracting the long-term monthly averages during a month of the rainy season is shown in the exceedance probability plots at the stations' locations (see Fig. ). Also, in Fig.  the limitation of this type of heterogeneity is observed during a dry month (July).

After downscaling, the heterogeneity is introduced back into the downscaled rainfall field. In this work, however, downscaling is performed beyond the largest resolution of TRMM (to reach a subkilometer rainfall resolution). Therefore, details about the local heterogeneity at the final downscaling resolution are required in order to account for spatial tendencies. Information at this scale is scarce or inexistent in the Andean high plateau; however, we rely on a rainfall field based on the stations' measurements (stations' locations are shown in Fig. ) and elevation maps widely used in hydrological, agricultural and other models in the area under study . These rainfall fields are generated using the thin-plate smoothing spline algorithm implemented in the ANUSPLIN 4.4 package for interpolation using scattered points of on-field measured rainfall together with latitude, longitude, and elevation as independent variables . It is worth mentioning that several references indicate its higher accuracy compared to other methods in several parts of the world . Also, the method has been widely applied on well-known and used climate products such as WorldClim (, http://www.worldclim.org) and IWMI (International Water Management Institute) Climate Atlas/CRU gridded data (, http://www.iwmi.org, http://www.cru.uea.ac.uk). The scalability properties of ANUSPLIN outputs are presented in Fig. . It is observed that the linearity of the plot of log⁡Mn(q) vs. log⁡λn is very good (determination coefficient is above 0.98 for all ANUSPLIN snapshots in the studied period from 1999 to 2006). Although these fields are “smooth” because of the usage of splines, these fields serve as a source for finding the monthly tendencies of rainfall at the desired subkilometer resolution. More specifically, the local heterogeneity G′ is the normalized field of spatial heterogeneities obtained as the monthly average of daily rainfall synthesized from ANUSPLIN output rainfall fields, which is equivalent to Eq. (). The inclusion of the local heterogeneity provides the corrected downscaled rainfall field R′ as follows: Ri,j′=KMi,jGi,j′∑i,jMi,jGi,j′, where K is the large-scale forcing scale factor preserving the rainfall mean magnitude . Finally, the heterogeneity for the months of February and August is shown in Fig.  for both TRMM and ANUSPLIN outputs.

To end this section, the lognormality of the corrected ANUSPLIN outcomes is tested (see Fig. ). That is, the monthly averages were pointwise filtered and then run over a lognormality test as it was performed on the TRMM data. Some disagreement is expected from the high rainfall intensities because of the northeast region being located in the rain forest rather than the Andean high plateau, and the fact that the ANUSPLIN outcomes did not use any station on that area for its construction. Note that since a lognormal distribution is stable, assembling all days in a period of time should amount to a set of numbers also lognormally distributed. Thus, Fig.  shows that the ANUSPLIN outcomes resemble a lognormal behavior for not too high rainfall intensities. Although, as already mentioned, it appears that the distribution may have a heavy tail for the reasons argued before.

Prior to applying a multifractal technique, an assessment of the available information was performed in order to detect multifractality in the data (TRMM precipitation in this case). This is summarized by the multifractal linear analysis . In summary, a multifractal characterization can only be performed if the log⁡–log⁡ plot of the sample moments versus the resolution parameter (Eqs. , ) shows a linear relationship for each moment order q (see Fig. ). The multifractal parameter τ is then obtained as the slope of the linear tendency for each q. It has been reported in and the possibility of two different multifractal behaviors at approximately the 16 km resolution (scale breaking). However, there is a fundamental difference between the data used in those studies and that used in this study. First, TRMM is obtained on a daily temporal framework and it is the result of a 3 h accumulation and extrapolated to reflect daily rainfall, whereas in and the temporal framework is on a minute scale. Therefore, if such scale breaking were to occur, the described temporal averaging or dressing of the TRMM data accounts for the hindering of such scale breaking. Figure  shows the difference of τ as a function of the moments q for the summer and winter seasons for the years 2001 and 2004. Each curve τ(q) corresponds to a nonzero rain day in the corresponding month.

The multifractal linear analysis was performed on TRMM data and also on the ANUSPLIN output snapshots. The latter was done in order to assess the validity of the equations presented in Sect.  when downscaling beyond the highest resolution of TRMM data. For TRMM data, three cascade levels were assessed in which the resolutions run from ∼ 225 to ∼ 28 km. For ANUSPLIN data, eight cascade levels were considered sweeping from ∼ 225 to ∼ 0.875 km. The cascade levels correspond to the scale parameter λn = 2n. The range of q's considered in the analysis was [0, 5]; see Fig.  for the analysis on 26 February 1999 for both TRMM and ANUSPLIN data. Specifically, the coefficients of determination for the linear regression for q's in the interval of [0, 5] are above 0.98 throughout the period of time studied (every day from 1999 to 2006). This result allows for the application of a multifractal procedure; i.e., the parameter τ is well defined in this range and therefore the MKP function can be used to infer the parameters β and σ. Here we have used finite difference methods to evaluate the first and second derivatives of τ(q), with respect to q as implemented by , , and . In particular, these derivatives were estimated using a partition of step 0.1 on the q values, so that τ′(1) and τ′′(1) were estimated using q = 0.9 and q = 1.1.

Lognormal parameters β and σ were computed for each snapshot (daily timescale). Their features, in this paper's application, correspond to specific aspects of precipitation dynamics. For example, β is associated with the degree of intermittency in the precipitation field, and σ determines the variance of the cascade generator. The relationship between the given information (rainfall intensity, variance, energy, entropy indicators, etc.) and the lognormal parameters is key to understand the role of the cascade parameters and the measurements. Here only the rainfall intensity (mean over all TRMM grids at time t) is compared with respect to the β and σ lognormal parameters. This relationship between data and the lognormal parameters is inferred from Eqs. (), () and (). In Sect. , τ was computed from the sample moments, which are direct consequence of the rainfall intensity data in Eq. (). Given that β and σ are obtained from Eq. (), they are implicitly related to τ. The monthly values of β and σ vs. the logarithm of mean rainfall intensity (per snapshot), for the period running from 1999 to 2006, are fitted with respect to the empirical models used in and . The nonlinear regressions for the months of February and July are provided in Fig. , and only snapshots with nonzero rainfall intensity were considered. These plots visually express the relationship of the parameters β and σ with respect to the large-scale forcing, which here is the mean. One can observe that while β shows high variability the parameter σ does not. In , σ was set to a fixed number due to its lack of sensitivity with respect to rainfall intensity. A similar behavior is observed in the Andean high plateau, but in this paper σ is allowed to vary. Given that the probability of nonzero rain is b-β, one can observe that in February the majority of β values are close to zero (probability close to 1), which is expected from a rainy month, whereas for July the probability of rain for snapshots with rain on them decreases to around 50 % indicated by the value of β ≈ 0.4. Note that the scatter plot for July is much less populated than the one for February since there are less rainy days during July. The stability analyses of β and σ are given in Tables  and . It is observed that the minimum value of the average β occurs in February (β = 0.1568 gives probability ≈ 80 %) and the maximum occurs in June (β = 0.7284 gives probability ≈ 35 %), as expected from the usual seasonal cycle of precipitation in the Andes. On the contrary, the exact opposite happens with the values of σ in which the wet season has more variance during the wet season compared to the localized rainfall during the dry season given by low values of σ. This last fact simply tells us that the rainfall is much more spread out in the region during the rainy season while it is more localized in the dry season. This is of course observed when comparing February and July in Fig. .

The main tool in the generation of rainfall data at smaller scales, as described in previous sections, is a disaggregation procedure having a random generator associated with a β-lognormal model with parameters β and σ. Recall that this generator includes the possibility of no rain in the generator (see Eq. ). Figure  shows TRMM data with no heterogeneity along with its downscaling to a subkilometer resolution averaging 1000 realizations and its heterogeneity correction as described in Sect.  . Specifically, the parameters of the β-lognormal distribution for this day are β = 0.0266 and σ = 0.1109.

Based on this consideration, the objective of this section is to compare the quality of the disaggregation procedure with respect to TRMM precipitation. The information to be generated is the mean rainfall intensity of all days in a month over the period of 8 years. Figure  shows two scatter plots comparing the means of observed and generated rainfall snapshots. The plots for February and July were chosen since they correspond to distinct rainfall patterns. The former is representative of a summer month (rainy season) and the latter is representative of a winter month (dry season). The scatter plots show all information over the period of 8 years to assure a good statistical assessment. Similar behavior is observed for all the other months (not shown). One can observe that the mean of each snapshot is preserved accurately as expected. This is caused by the large-scale forcing factor, K, used in the correction procedure described in Sect. . On the other hand, increasing the resolution via the downscaling process increases the variability (or randomness) of rainfall as expected from the fact that the sample q = 2 moment is proportional to λnτ(2). This is observed when comparing the variances of observed and generated snapshots over a period of time. In addition, the heterogeneity correction procedure was devised for preserving the means and not other statistical moments. Also, note that the increment in variability changes depending on the season, i.e., February (wet season) shows on average about 4 times more variability after downscaling and correction (values outside the regression line). Observe that the variance can only increase, and that the very few cases in Fig.  were the variance decreases are mainly due to the correction process. On the other hand, July (dry season) exhibits about 2.5 times more variability after downscaling. In general, an argument for such changes in variance is that, from a temporal point of view, the second moments are proportional to λnτ(2) in contrast with the case for q = 1, where all τ(q) curves coincide (τ(1) = 0 always) (see Fig. ).

With the objective of illustrating the disaggregation procedure used in the paper, Fig.  presents the level by level downscaling for 25 January 2000.

Also, the ensemble mean realization is given in Fig.  for 10, 100 and 1000 realizations of the downscaling and correction procedure described in Sect. .

This section begins with a pointwise (spatial) comparison of some statistics between observed and generated (downscaled and corrected) time series for the Capachica and Cojata stations. Such time series are shown in Fig. . In Table  the statistics of observed and generated time series are provided. In particular, it is observed that the Hurst exponents for the whole 8 years (all seasons) between on-site observed and generated (downscaled and corrected) rainfall mostly agree. For the Capachica station, it was found that Hobs = 0.7453 and Hgen = 0.6750; for the Muñani station Hobs = 0.6766 and Hgen = 0.6838. A Hurst exponent in the range 0.5 < H < 1 indicates a long-term positive autocorrelation, which implies the tendency of a high value to be followed by another high value. This behavior, H > 0.5, is shared by the other stations as well. However, the fact that H ≠ 0.5 may indicate that the multifractal field is not conservative, which is usually handled by studying the field fluctuations . However, H here is in general close to 0.5 and the differences can be attributed to, for example, uncertainty/error in the precipitation measurements of both TRMM and the stations. A fluctuation analysis would be the concern of future research. Also, it is important to highlight that the only source of correlation among snapshots of downscaled rainfall is carried over from the correlation of TRMM data since the downscaling process is exclusively spatial.